Introduction:

 Here we will talk about details of nuclear reactions in which neutrons are projectiles. A good source of neutrons is  α-particles bombarding light elements.

⁴He₂ + ⁹Be₄ → ¹²C + ¹ n ₀ + Q

The α-particles are generally achieved from radium (Ra) in regular radioactive procedures. In such type of a reaction, up to 10⁷ neutrons are ejected by means of energy 1-13MeV. Neutrons with this category of energy are referred as fast neutrons, for the reason of nuclear fission and large scale discharge of atomic energy; neutrons of energy in the neighborhood of 0.0025eV are desired. The neutrons are termed as thermal neutrons, Due to this is regarding the thermal energy 3 ⁄ 2 KT of molecules at thermal temperature. The method of slow down fast neutrons is known as "Thermalisation or Moderation".

Neutron Moderation:

The process of receiving fast neutrons to leisurely down to thermal neutron is considered as moderation or thermalisation. This is obtained by passing the fast neutrons by some suitable material termed as moderators. In like this way that neutrons are not loosed through absorption, but merely have their kinetic energy decreased gradually by elastic collision with the material's nuclei , such as good moderators are graphite and water. There are usually two frames of reference in the study of the dynamics of neutrons.

Laboratory frame:

In this frame, the target nucleus is at relaxed earlier than collision and it is reached by a projectile neutron with velocity Vo. After collision, nucleus is spread by angle θ and the target nucleus turns by angle θ′.

The Centre Of Mass Frame:

This is theoretical but a extremely helpful technique in dealing with dynamics of moderation process. In this frame, the centre of mass of neutron and target nucleus is at rest and it is approached in reverse direction by the neutron and target nucleus. By principle of conservation of momentum,

mV_c - MV = 0

or V_c = MV ⁄ m

Here V_c is velocity of neutron and V is velocity of target nucleus. After collision, neutron scatters off with angle Ø and nucleus pass all via (Ø + π ). To conserve momentum, velocity after collision remains unaffected.

XY = QZ

Sinθ = QZ / V₁

V ₁ sinθ = QZ

V_c Sin Ø  = XY

V₁ ⁄V_c=sin Ø /sinθ

In laboratory framework, relative velocity = Vo as M is at rest. In centered mass frame, relative velocity = Vc + V while they move toward reverse direction.    

These two relative velocities have to be equal in the 2 frames, in other words,

V_o = V_c + V

Combing equations and eliminating V_c

V = mV₀/(M + m)

And V_c = MV₀/(M + m) (eliminating V)

m/M ≈ 1/A

Here A is the mass number of nucleus. Therefore:

V = V₀/1 + A

V₀ = V(1 + A)

Such that V_c = V₀A/1+A and V_c/V = V₀A/1+A . 1+A/V₀

V₁² = V² + V_c² - 2VV_Ccos(180⁰ - Φ)

= V² + V_c² - 2VV_C cosΦ

= V² [1 + V_c²/V² + 2V_ccosΦ/V]

V₁² = V²(1 + A² + 2AcosΦ)

So,

Let the incident energy of the neutron from the laboratory frame be given as:

E₀ = 1/2mV₀²

After first collision, energy is reduced to:

E₁ = = 1/2mV₁²

And fractional energy is given as:

E₁/E₂ = 1/2MV₁²/1/2MV₀² = V₁²/V₀²

E₁/E₀ = (1 + A² + 2AcosΦ)/(1 + A)² (Fractional energy)

Cases of Interest:

1. Glancing of collision; (Φ = 0)

E₁/E₀ = (1 + A² + 2AcosΦ)/(1 + A)² = 1 + A² +2A/1 + A² + 2A = 1

This signifies E_{1 ≡} E₀ thus neutron loses little or no energy on colliding with nucleus.

2.  Head on collision,( Φ =π)  Where, Cos π = - 1

 E₁/E_{0 =} (1+A²-2A)/(1+A)² = (A-1)²/(A+1)²

This kind of collision causes maximum energy loose through neutron such as for graphite that is a carbon allotrope and with its atomic mass, A = 12.   E₁/E₀ = 72 %

3.General case: 0 < Φ < π

Introducing α = [A-1/A+1]2

(ΔE)max = (E₀ - E1)max

= E₀(1 - E₁/E₀)

= E₀(1 - α)

= (ΔE/E₀)_max = (1- α)

= 1-[A-1/A+1]2

= 4/A - 8/A² + 12/A³ - 16/A⁴ + 20/A⁵

(ΔE/E₀)_max = 1-(1-1/A)²/(1+1/A)²

4/A[1-2/A + 3/A² - 4/A³ + 5/A⁴]

A very significant point is that for material to act as good moderator, fractional energy loss should be large and from expression above, loss is smaller. From expression given above, it is seen that fractional energy loss (ΔE/E₀)_max is inversely proportional to atomic mass of moderator.

Passage of a Beam of a Neutron through a Moderating Material:

A flux of up to 10⁸ fast neutrons to be moderated by several nuclei of moderator. The probability that single neutron will be scattered through angle Φ lying between Φ and dΦ such that energy of the neutron after scattering lies between E + dE. Range of energy is E₁ = E_o for glancing angle collision and E₁ = αE₀ for head on collision. Probability that neutron will have energy E where αE₀ is less than E and less than E_o(αE₀<E<E₀) after an arbitrary scattering is P(E). Energy between this range is:

E-αE₀ = E₀(1-α) so that P(E)dE = dE/E₀(1-α)

Normalizing this probability

_E1=α∫ ^E0E₀P(E)dE = _αE0∫^E0(dE/E₀(1-α)) = 1

Average energy of neutron after series of scattering is probability that single collision will have energy E. Then, average energy is provided by:

<E> = [1/2[E₀(1-α)][E²]_αE0^E0

<E> = 1/2[E₀(1-α)

Where α = (A-1)²/(A+1)²

Average log energy decrement (ξ):

This term is introduced to get information about average number of collision which a fast neutron will make before its energy E_o is reduced to thermal energy E_t, when E_o is reduced to E, log energy decrement is provided by:

log_eE - logeE => loge(E₀/E)

Average log decrement = <log_e(E₀/E)>

ζ = _αE0∫^E0 log_e(E₀/E)P(E)/dE

As _αE0∫^E0P(E)/dE = 1

Take x = E/E₀

Then for E = αE₀ where x = α

ζ = -1/1-α∫_α¹logxdx

= 1+ [α/1-α](logα)

By substituting α = (A-1/A+1)²

ζ = 1-(A-1)²/2A loge(A-1/A+1)

For A > 1, a convenient approximation

ζ = 2/A + 2/3

ζ α 1/A => n = 1/ζ loge(E₀/E)

Where n is the number of collision needed to reduce fast neutron energy E_o to thermal energy E_t.

Slowing down power Sp:

Effectiveness of moderating material is not only by log energy decrement but also by density of substance, i.e., number of colliding centers which a material has per unit volume.

Material should also have small absorption cross section, σ_a. Slowing down power is term that combines different parameters and it is expressed as:

S_p = ξNσ_a Or S_p = ξN_aρσ_a/A

Where ρ is the density

A is atomic weight of moderator

σ_a = Absorption cross section

N_a = Avogadro's constant

Neutron interaction:

All neutrons at the time of their birth are fast. In penetrating through matter, they suffer characteristics procedure of energy degradation or moderation. Probability of neutron interaction with the nucleus happening in moderating medium is cross-section signified by σ. This is estimated as effective area presented to neutron and it is defined in unit of barns.

Where 1 barn = 1 x 10^-24cm² (total part of material presented for interaction).

The total cross-section σt has several components:

σ_t = σ_et + σ_inelastic + σ_ab + σ_f + ......

This is a sum of elastic, inelastic, absorption and fusion. All cross-sections are strongly energy dependent. The total cross-section μ_t is a microscopic quality. When multiplied by number N of absorber atoms per unit volume we have

Σ=σN

The removal of neutrons from a beam transversing a thickness t

I = Ioe^-σNt Or I = I₀e^Σt

I_o - initial intensity of beam neutron

Mean Free Path is distance travelled by neutron before collision

π = 1/Σ or 1/σN

Absorption:

As neutrons near thermal energy, the probability of capture by absorber nucleus increases i.e. absorption cross-section increases σ_ab. For several absorbing nucleic, as neutron energy become very small (0.01 to 10,000eV), absorption cross-section is directly proportional to inverse of velocity and inversely proportional to energy.

σ_ab α 1/v + 1/√E

Between range of energy (0.001 to 10,000eV) if the neutron enters reactor or start in reactor with certain energy E_o and having energy E after certain number of collision.

σ₀/σ = √E/√E₀

From absorption spectrum of neutrons inside reactor, resonance effect is observed.

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